Simple Harmonic Motion (SHM)

Describing Oscillation

Simple harmonic motion (SHM) is a type of periodic motion where an object moves back and forth about an equilibrium position under a restoring force proportional to its displacement. A classic example is a mass attached to a spring oscillating on a frictionless surface.

• Equilibrium Position: The point where the net force on the object is zero.

• Restoring Force: The force that brings the object back toward equilibrium, typically given by Hooke's Law: .

• Displacement (): The distance from the equilibrium position at any instant.

• Oscillation: The repetitive movement of the object through its equilibrium position.

Amplitude, Period, Frequency, and Angular Frequency

Several key quantities describe SHM:

• Amplitude (): The maximum displacement from equilibrium.

• Period (): The time taken for one complete oscillation (cycle).

• Frequency (): The number of oscillations per second.

• Angular Frequency (): (measured in radians per second).

Mathematical Formulation of SHM

SHM occurs when the restoring force is directly proportional to the displacement and acts in the opposite direction:

• Hooke's Law:

• Equation of Motion:

• General Solution: , where is the phase constant.

• Angular Frequency:

Circular Motion and SHM

SHM can be visualized as the projection of uniform circular motion onto a diameter. If a point moves in a circle of radius with constant angular speed , its projection onto a diameter executes SHM.

• Position:

• Acceleration:

• Relationship: The acceleration is always directed toward the equilibrium position and is proportional to the displacement.

Displacement, Velocity, and Acceleration in SHM

The position, velocity, and acceleration as functions of time are:

• Displacement:

• Velocity:

• Acceleration:

The velocity is zero at maximum displacement, and maximum at the equilibrium position. Acceleration is maximum at maximum displacement and zero at equilibrium.

Energy in SHM

In ideal SHM (no friction), the total mechanical energy is conserved and is the sum of kinetic and potential energies:

• Total Energy: (constant)

• Kinetic Energy:

• Potential Energy:

• Maximum Velocity:

Worked Examples

Several worked examples illustrate the application of SHM equations to solve for displacement, velocity, acceleration, and energy at various points in the motion. These include:

• Finding the time when a block reaches a certain displacement.

• Calculating the speed and acceleration at a given position.

• Analyzing the direction of motion and acceleration based on the sign of velocity and acceleration.

The Simple Pendulum

Physical Model and Restoring Force

A simple pendulum consists of a point mass suspended from a string of length . When displaced from equilibrium, it oscillates under the influence of gravity. For small angles, the restoring force is proportional to the displacement along the arc, leading to SHM.

• Restoring Force: (for small in radians)

• Equation of Motion:

• Angular Frequency:

• Period:

For larger angles, the motion is not strictly SHM, and a more general expression for the period must be used:

• General Period Expression:

Example: Error in Small Angle Approximation

When the amplitude of oscillation is not small, the period calculated using the small angle approximation can differ from the true period. The error becomes significant for angles greater than about (as shown in the worked example).

Summary Table: Key Quantities in SHM and the Simple Pendulum

Key Takeaways

• SHM is characterized by a restoring force proportional to displacement and directed toward equilibrium.

• The motion can be described mathematically using sine and cosine functions.

• Energy in SHM oscillates between kinetic and potential forms, but the total energy remains constant (in the absence of friction).

• The simple pendulum approximates SHM for small angular displacements.

• For larger amplitudes, corrections to the period must be made.

Additional info: These notes include both typed and handwritten worked examples, reinforcing the application of SHM concepts to practical problems. The summary table provides a quick reference for comparing the mass-spring system and the simple pendulum.